Utilitarian cake-cutting is a rule for dividing a heterogeneous resource, such as a cake or a land-estate, among several partners with different cardinal utility functions. The rule says that the cake should be divided in a way which maximizes the sum of the utilities of the partners. It is called "utilitarian" because it is inspired by utilitarianism. An alternative term is maxsum cake-cutting.
Contents
- Example
- Notation
- Utilitarianism and Pareto efficiency
- Characterization of the utilitarian rule
- Disconnected pieces
- Connected pieces
- Utilitiarianism vs fairness
- References
Utilitarian cake-cutting is often not "fair"; hence, utilitarianism is in conflict with fair cake-cutting.
Example
Consider a cake with two parts: chocolate and vanilla, and two partners: Alice and George, with the following valuations:
The utilitarian rule gives each part to the partner with the highest utility. In this case, the utilitarian rule gives the entire chocolate to Alice and the entire Vanilla to George. The maxsum is 13.
The utilitarian division is not fair: it is not proportional since George receives less than half the total cake value, and it is not envy-free since George envies Alice.
Notation
The cake is called
There are
A division
The concept is often generalized by assigning a different weight to each partner. A division
where the
Utilitarianism and Pareto-efficiency
Every WUM division with positive weights is obviously Pareto-efficient. This is because, if a division
What's more surprising is that every Pareto-efficient division is WUM for some selection of weights.
Characterization of the utilitarian rule
Christopher P. Chambers suggests a characterization to the WUM rule. The characterization is based on the following properties of a division rule R:
The following is proved for partners that assign positive utility to every piece of cake with positive size:
Disconnected pieces
When the value functions are additive, utilitarian-maximal divisions always exist. Intuitively, we can give each fraction of the cake to the partner that values it the most, as in the example above. Similarly, WUM divisions can be found by giving each fraction of the cake to the partner for whom the ratio
This process is easy to carry out when the value functions are piecewise-constant, i.e. the cake can be divided to a finite number of pieces such that the value density of each piece is constant for all people.
What happens when the value functions are general (not piecewise-constant)? Then, the simple procedure of "give each piece to the one which wants it the most" does not work because there is an infinite number of different "pieces" to consider.
The good news is that UM divisions still exist. This is a corollary of the Dubins–Spanier compactness theorem and it can also be proved using the Radon–Nikodym set.
The bad news is that no finite algorithm can find a UM division. Proof: A finite algorithm has value-data only about a finite number of pieces. I.e. there is only a finite number of subsets of the cake, for which the algorithm knows the valuations of the partners. Suppose the algorithm has stopped after having value-data about
Connected pieces
When the cake is 1-dimensional and the pieces must be connected, the simple algorithm of assigning each piece to the agent that values it the most no longer works, even when the pieces are piecewise-constant. In this case, the problem of finding a UM division is NP-hard, and furthermore no FPTAS is possible unless P=NP.
There is an 8-factor approximation algorithm, and a fixed-parameter tractable algorithm which is exponential in the number of players.
For every set of positive weights, a WUM division exists and can be found in a similar way.
Utilitiarianism vs. fairness
A utilitarian division is not always fair; see the #Example. Similarly, a fair division is not always utilitarian-maximal. Fairness comes with a price-tag: the price of fairness is the amount of welfare that society has to "pay" in order to maintain the ideal of fairness.
One approach to combining utilitarianism and fairness is to find, among all possible fair divisions, a fair division with a highest sum-of-utilities. In particular, the following algorithm is used to find an envy-free cake-cutting with maximum sum-of-utilities, for a cake which is a 1-dimensional interval, when each person may receive disconnected pieces and the value functions are additive:
- For
n partners with piecewise-constant valuations: create a set of totally constant pieces. solve a set of linear equations. Give each partner a fraction of each totally constant piece. - For
2 partners with piecewise-linear valuations: for each point in the cake, calculate the ratio between the utilities:r = u 1 / u 2 r ≥ r ∗ r < r ∗ r ∗ r ∗ r ∗ - For
n partners with general valuations: additive approximation to envy and efficiency, based on the piecewise-constant-valuations algorithm.
See for further discussion of these results relating them to equitable division.